Book Review
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MathSciNet review:
1568098
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Book Information:
Author:
S. Franciosi B. Amberg, and F. de Giovanni
Title:
Products of groups
Additional book information:
Oxford Mathematical Monographs, Clarendon Press, Oxford, 1992, xii+220 pp., US$78.00. ISBN 0-19-853575-9.
Bernhard Amberg, Abelian factorizations of infinite groups, Math. Z. 123 (1971), 201–214. MR 289643, DOI 10.1007/BF01114789
Bernhard Amberg, Silvana Franciosi, and Francesco de Giovanni, On normal subgroups of products of nilpotent groups, J. Austral. Math. Soc. Ser. A 44 (1988), no. 2, 275–286. MR 922613
Bernhard Amberg, Silvana Franciosi, and Francesco de Giovanni, Groups with a nilpotent triple factorisation, Bull. Austral. Math. Soc. 37 (1988), no. 1, 69–79. MR 926977, DOI 10.1017/S0004972700004159
Bernhard Amberg, Silvana Franciosi, and Francesco de Giovanni, On trifactorized soluble minimax groups, Arch. Math. (Basel) 51 (1988), no. 1, 13–19. MR 954063, DOI 10.1007/BF01194149
Bernhard Amberg, Silvana Franciosi, and Francesco de Giovanni, Triply factorized groups, Comm. Algebra 18 (1990), no. 3, 789–809. MR 1052766, DOI 10.1080/00927879008823944
Christopher J. B. Brookes, Ideals in group rings of soluble groups of finite rank, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 1, 27–49. MR 764490, DOI 10.1017/S0305004100062551
N. S. Černikov, On the product of almost abelian groups, Ukrain. Mat. Zh. 33 (1981), no. 1, 136–138 (Russian). MR 605475
N. S. Chernikov, Products of groups of finite free rank, Groups and systems of their subgroups, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1983, pp. 42–56 (Russian). MR 763879
P. M. Cohn, A remark on the general product of two infinite cyclic groups, Arch. Math. (Basel) 7 (1956), 94–99. MR 81278, DOI 10.1007/BF01899562
Silvana Franciosi and Francesco de Giovanni, On trifactorized soluble groups of finite rank, Geom. Dedicata 38 (1991), no. 3, 331–341. MR 1112670, DOI 10.1007/BF00181195
J. D. Gillam, A finite $p$-group $P=AB$ with $\textrm {Core}(A)=\textrm {Core}(B)=\{1\}$, Rocky Mountain J. Math. 3 (1973), 15–17. MR 347971, DOI 10.1216/RMJ-1973-3-1-15
H. Heineken and J. C. Lennox, A note on products of abelian groups, Arch. Math. (Basel) 41 (1983), no. 6, 498–501. MR 731631, DOI 10.1007/BF01198577
H. Heineken and I. J. Mohamed, A group with trivial centre satisfying the normalizer condition, J. Algebra 10 (1968), 368–376. MR 235035, DOI 10.1016/0021-8693(68)90086-0
D. F. Holt and Robert B. Howlett, On groups which are the product of two abelian groups, J. London Math. Soc. (2) 29 (1984), no. 3, 453–461. MR 754932, DOI 10.1112/jlms/s2-29.3.453
Robert B. Howlett, On the exponent of certain factorizable groups, J. London Math. Soc. (2) 31 (1985), no. 2, 265–271. MR 809947, DOI 10.1112/jlms/s2-31.2.265
Bertram Huppert, Über das Produkt von paarweise vertauschbaren zyklischen Gruppen, Math. Z. 58 (1953), 243–264 (German). MR 55341, DOI 10.1007/BF01174144
Noboru Itô, Über das Produkt von zwei abelschen Gruppen, Math. Z. 62 (1955), 400–401 (German). MR 71426, DOI 10.1007/BF01180647
[18] S. V. Ivanov, On general products of groups, Proc. All-Union Sympos. Group Theory, Sverdlovsk, 1989, pp. 147-148.
Otto H. Kegel, Produkte nilpotenter Gruppen, Arch. Math. (Basel) 12 (1961), 90–93 (German). MR 133365, DOI 10.1007/BF01650529
Otto H. Kegel, Zur Struktur mehrfach faktorisierter endlicher Gruppen, Math. Z. 87 (1965), 42–48 (German). MR 170938, DOI 10.1007/BF01109929
A. G. Kurosh, The theory of groups, Chelsea Publishing Co., New York, 1960. Translated from the Russian and edited by K. A. Hirsch. 2nd English ed. 2 volumes. MR 0109842
John C. Lennox and Derek J. S. Robinson, Soluble products of nilpotent groups, Rend. Sem. Mat. Univ. Padova 62 (1980), 261–280. MR 582956
John C. Lennox and James E. Roseblade, Soluble products of polycyclic groups, Math. Z. 170 (1980), no. 2, 153–154. MR 562584, DOI 10.1007/BF01214770
Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl, The maximal factorizations of the finite simple groups and their automorphism groups, Mem. Amer. Math. Soc. 86 (1990), no. 432, iv+151. MR 1016353, DOI 10.1090/memo/0432
[25] R. Maier, Um problema da teoria dos subgrupos subnomais, Bol. Soc. Brasil. Mat. 8 (1977), 127-130.
[26] O. Ore, Contributions to the theory of groups of finite orders, Duke. Math. J. 5 (1938), 431-460.
L. Rédei, Zur Theorie der faktorisierbaren Gruppen. I, Acta Math. Acad. Sci. Hungar. 1 (1950), 74–98 (German, with Russian summary). MR 48435, DOI 10.1007/BF02022554
Derek J. S. Robinson and Stewart E. Stonehewer, Triple factorizations by abelian groups, Arch. Math. (Basel) 60 (1993), no. 3, 223–232. MR 1201635, DOI 10.1007/BF01198805
J. E. Roseblade, Group rings of polycyclic groups, J. Pure Appl. Algebra 3 (1973), 307–328. MR 332944, DOI 10.1016/0022-4049(73)90034-0
N. F. Sesekin, The product of finitely connected abelian groups, Sibirsk. Mat. . 9 (1968), 1427–1430 (Russian). MR 0236262
N. M. Suchkov, An example of a mixed group factorable by two periodic subgroups, Algebra i Logika 23 (1984), no. 5, 573–577, 600 (Russian). MR 817031
Ya. P. Sysak, Products of infinite groups, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 53 (1982), 36 (Russian). MR 705793
Ya. P. Sysak, Radical modules over groups of finite rank, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 18 (1989), 51 (Russian). MR 1046494
[34] J. Szép, On factorizable non-simple groups, Acta Univ. Szeged. Sect. Sci. Math. 13 (1950), 239-241.
M. J. Tomkinson, Products of abelian subgroups, Arch. Math. (Basel) 47 (1986), no. 2, 107–112. MR 859258, DOI 10.1007/BF01193678
Helmut Wielandt, Über Produkte von nilpotenten Gruppen, Illinois J. Math. 2 (1958), 611–618 (German). MR 136656
Helmut Wielandt, Subnormalität in faktorisierten endlichen Grupppen, J. Algebra 69 (1981), no. 2, 305–311 (German). MR 617081, DOI 10.1016/0021-8693(81)90207-6
John S. Wilson, Soluble groups which are products of groups of finite rank, J. London Math. Soc. (2) 40 (1989), no. 3, 405–419. MR 1053611, DOI 10.1112/jlms/s2-40.3.405
D. Ī. Zaĭtsev, Nilpotent approximations of metabelian groups, Algebra i Logika 20 (1981), no. 6, 638–653, 728 (Russian). MR 675122
Guido Zappa, Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili tra loro, Atti Secondo Congresso Un. Mat. Ital., Bologna, 1940, Edizioni Cremonese, Rome, 1942, pp. 119–125 (Italian). MR 0019090
- [1]
- B. Amberg, Abelian factorizations of infinite groups, Math. Z. 123 (1971), 201-214. MR 0289643 (44:6831)
- [2]
- B. Amberg, S. Franciosi, and F. de Giovanni, On normal subgroups of products of nilpotent groups, J. Austral. Math. Soc. Series A 44 (1988), 275-286. MR 922613 (88m:20078)
- [3]
- -, Groups with a nilpotent triple factorization, Bull. Austral. Math. Soc. 37 (1988), 69-79. MR 926977 (89g:20062)
- [4]
- -, On trifactorized soluble minimax groups, Arch. Math. (Basel) 51 (1988), 13-19. MR 954063 (89g:20061)
- [5]
- -, Triply factorized groups, Comm. Algebra 18 (1990), 789-809. MR 1052766 (91e:20029)
- [6]
- C. J. B. Brookes, Ideals in group rings of soluble groups of finite rank, Math. Proc. Cambridge Philos. Soc. 97 (1985), 27-49. MR 764490 (86j:16014)
- [7]
- S. N. Černikov, Products of almost abelian groups, Ukrain. Mat. Ž. 33 (1981), 136-138. MR 605475 (82c:20058)
- [8]
- -, Products of groups of finite free rank, Groups and Systems of their Subgroups, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1983, pp. 42-56. MR 763879 (86d:20024)
- [9]
- P. M. Cohn, A remark on the general product of two infinite cyclic groups, Arch. Math. (Basel) 7 (1956), 94-99. MR 0081278 (18:376e)
- [10]
- S. Franciosi and F. de Giovanni, On trifactorized soluble groups of finite rank, Geom. Dedicata 38 (1991), 331-341. MR 1112670 (92f:20033)
- [11]
- J. D. Gillam, A finite p-group with , Rocky Mountain J.Math. 3 (1973), 15-17. MR 0347971 (50:469)
- [12]
- H. Heineken and J. C. Lennox, A note on products of abelian groups, Arch. Math. (Basel) 41 (1983), 498-501. MR 731631 (86b:20061)
- [13]
- H. Heineken and I. J. Mohamed, A group with trivial centre satisfying the normalizer condition, J. Algebra 10 (1968), 368-376. MR 0235035 (38:3347)
- [14]
- D. F. Holt and R. B. Howlett, On groups which are the product of two abelian groups, J. London Math. Soc. (2) 29 (1984), 453-461. MR 754932 (86f:20028)
- [15]
- R. B. Howlett, On the exponent of certain factorizable groups, J. London Math. Soc. (2) 31 (1985), 265-271. MR 809947 (87b:20040)
- [16]
- B. Huppert, Über das Produkt von paarweise vertauschbaren zyklischen Gruppen, Math. Z. 58 (1954), 243-264. MR 0055341 (14:1059f)
- [17]
- N. Itô, Über das Produkt von zwei abelschen Gruppen, Math. Z. 62 (1955), 400-401. MR 0071426 (17:125b)
- [18]
- S. V. Ivanov, On general products of groups, Proc. All-Union Sympos. Group Theory, Sverdlovsk, 1989, pp. 147-148.
- [19]
- O. H. Kegel, Produkte nilpotenter Gruppen, Arch. Math. (Basel) 12 (1961), 90-93. MR 0133365 (24:A3199)
- [20]
- -, Zur Struktur mehrfach faktorisierter endlicher Gruppen, Math. Z. 87 (1965), 42-48. MR 0170938 (30:1173)
- [21]
- A. G Kuroš, The theory of groups, 2nd ed., Chelsea, New York, 1960. MR 0109842 (22:727)
- [22]
- J. C. Lennox and D. J. S. Robinson, Soluble products of nilpotent groups, Rend. Sem. Mat. Univ. Padova 62 (1980), 261-280. MR 582956 (81j:20050)
- [23]
- J. C. Lennox and J. E. Roseblade, Soluble products of polycyclic groups, Math. Z. 170 (1980), 153-154. MR 562584 (81a:20047)
- [24]
- M. W. Liebeck, C. E. Praeger, and J. Saxl, The maximal factorizations of the finite simple groups and their automorphism groups, Mem. Amer. Math. Soc., vol. 432, Amer. Math. Soc., Providence, RI, 1990. MR 1016353 (90k:20048)
- [25]
- R. Maier, Um problema da teoria dos subgrupos subnomais, Bol. Soc. Brasil. Mat. 8 (1977), 127-130.
- [26]
- O. Ore, Contributions to the theory of groups of finite orders, Duke. Math. J. 5 (1938), 431-460.
- [27]
- L. Rédei, Zur Theorie der faktorisierbaren Gruppen. I, Acta. Math. Hungar. 1 (1950), 74-98. MR 0048435 (14:13f)
- [28]
- D. J. S. Robinson and S. E. Stonehewer, Triple factorizations by abelian groups, Arch. Math. (Basel) 60 (1993), 223-232. MR 1201635 (94d:20034)
- [29]
- J. E. Roseblade, Group rings of polycyclic groups, J. Pure Appl. Algebra 3 (1973), 307-328. MR 0332944 (48:11269)
- [30]
- N. F. Sesekin, Products of finitely connected abelian groups, Sibirsk. Mat. Zh. 9 (1968), 1427-1430. MR 0236262 (38:4559)
- [31]
- N. M. Suchkov, Example of a mixed group factorized by two periodic subgroups, Algebra i Logika 23 (1984), 573-577. MR 817031 (87d:20058)
- [32]
- Sysak, Y. P., Products of infinite groups, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 82.53 (1982). MR 705793 (84m:20032)
- [33]
- -, Radical modules over groups of finite rank, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 89.18 (1989). MR 1046494 (91b:20023)
- [34]
- J. Szép, On factorizable non-simple groups, Acta Univ. Szeged. Sect. Sci. Math. 13 (1950), 239-241.
- [35]
- M. J. Tomkinson, Products of abelian subgroups, Arch. Math. (Basel) 47 (1986), 107-112. MR 859258 (87i:20063)
- [36]
- H. Wielandt, Über Produkte von nilpotenten Gruppen, Illinois J. Math. 2 (1958), 611-618. MR 0136656 (25:121)
- [37]
- -, Subnormalität in faktorisierten endlichen Gruppen, J. Algebra 69 (1981), 305-311. MR 617081 (82i:20038)
- [38]
- J. S. Wilson, Soluble groups which are products of groups of finite rank, J. London Math. Soc. (2) 40 (1989), 405-419. MR 1053611 (91b:20044)
- [39]
- D. I. Zaĭcev, Nilpotent approximations of metabelian groups, Algebra i Logika 20 (1981), 638-653. MR 675122 (84b:20045)
- [40]
- G. Zappa, Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili tra loro, Atti del Secondo Congresso dell'Unione Matematica Italiana, Cremonese, Rome, 1940, pp. 119-125. MR 0019090 (8:367d)
Review Information:
Reviewer:
Derek J. S. Robinson
Journal:
Bull. Amer. Math. Soc.
30 (1994), 262-268
DOI:
https://doi.org/10.1090/S0273-0979-1994-00460-4